Victor Moses and Mame Biram Diouf scored the goals to move Stoke up to eighth and leave Everton just six points above the drop zone after a five-game winless streak in the Premier League.
The visitors saw their best chance fall to substitute Arouna Kone, who shot at Asmir Begovic, and were then jeered off by the Everton faithful present at the Britannia Stadium on Wednesday.
Martinez's side have progressed into the UEFA Europa League last 16, but it is their domestic form that has caused concern compared to last season when Everton finished fifth to qualify for Europe.
A downbeat Martinez said: "We carried such negative momentum that everything that could go wrong, it went wrong.
"It probably showed that it was our fifth game in two weeks.
"We are not looking down, that is the reality. We want to get as many points as we can and we have got 10 games to go.
"Six of those are at Goodison [Park] which is an important factor, to be able to have our fans with us.
"It's been a tough period and you can understand the frustration that we all share.
"We have got 30 points to fight for and we are going to give our lives to get as many as we can. With our support and everyone at the club, this is the moment to get together."
Martinez expects left-back Leighton Baines, who missed the Stoke game through injury, to return for Everton's next Premier League game against Newcastle United.
He added: "Maybe [next] Thursday [against Dynamo Kiev in the Europa League] is a bit too early but I am confident that he will be back soon."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.